applied sciences

Article

Prediction of Cone Crusher Performance ConsideringLiner WearYanjun Ma 1, Xiumin Fan 1,2,* and Qichang He 1

1 School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China;kfmyj@163.com (Y.M.); heqc@sjtu.edu.cn (Q.H.)

2 State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University,Shanghai 200240, China

* Correspondence: xmfan@sjtu.edu.cn; Tel.: +86-21-3420-6291

Academic Editor: César M. A. VasquesReceived: 26 September 2016; Accepted: 29 November 2016; Published: 3 December 2016

Abstract: Cone crushers are used in the aggregates and mining industries to crush rock material.The pressure on cone crusher liners is the key factor that influences the hydraulic pressure,power draw and liner wear. In order to dynamically analyze and calculate cone crusher performancealong with liner wear, a series of experiments are performed to obtain the crushed rock materialsamples from a crushing plant at different time intervals. In this study, piston die tests are carriedout and a model relating compression coefficient, compression ratio and particle size distributionto a corresponding pressure is presented. On this basis, a new wear prediction model is proposedcombining the empirical model for predicting liner wear with time parameter. A simple and practicalmodel, based on the wear model and interparticle breakage, is presented for calculating compressionratio of each crushing zone along with liner wear. Furthermore, the size distribution of the productis calculated based on existing size reduction process model. A method of analysis of productsize distribution and shape in the crushing process considering liner wear is proposed. Finally,the validity of the wear model is verified via testing. The result shows that there is a significantimprovement of the prediction of cone crusher performance considering liner wear as compared tothe previous model.

Keywords: cone crusher; wear; modelling; crushing; size reduction

1. Introduction

Cone crusher performance is usually defined by the terms particle size distribution (PSD),capacity and particle shape. In this paper, the crusher performance refers to PSD and particle shape.While the cone crusher is working, the mantle moves around the axis of the crushing chamber.As the mantle swings between the closed and open sides, the rock material is squeezed and crushedbetween two liners. When rock material is getting crushed in the crushing chamber, there is pressureon the liner, which leads to more serious liner wear over time. The geometry of crushing chamber iscrucial for the performance. Due to wear, the geometry of the liner changes, which causes the closeside setting (CSS) to gradually increase. Hence, cone crusher performance is affected by graduallydeteriorating performance over time [1–3]. This will affect the PSD curve and the flakiness indexcurve [4,5]. Thus, in order to compensate the working process of liner wear and enhance its workingperformance, it is meaningful for studying the impact of liner wear on cone crusher performance.

Evertsson [6,7] developed a flow model, a size reduction model and a pressure response model,which made it possible to model the behavior of a cone crusher [8–11]. Some researchers haveinvestigated the wear model of crushers [12]. Lindqvist and Evertsson [13] showed that the linerwear is proportional to the crushing pressure, which is mainly dependent on compression ratio and

Appl. Sci. 2016, 6, 404; doi:10.3390/app6120404 www.mdpi.com/journal/applsci

Appl. Sci. 2016, 6, 404 2 of 13

particle size distribution index. Rahul [14] performed experiments based on full factorial designadopted from statistical modelling software, and the input parameters like load, sliding distance,hardness of coal, and hardness of liner material are taken into consideration, and weight loss wasconsidered as the output response. Lindqvist and Evertsson [15] introduced the method of calculatingthe compression ratio and the pressure distribution angle of each crushing zone, which were keyparameters for calculating the distribution of the pressure on the liner. In addition, a wear modelwas developed by Lindqvist [16]. Recently, wear prediction using Discrete Element Method (DEM)was proposed [17] with a wear model to predict wear on the liner of a mill, obtaining very goodagreement using an abrasion measure as the wear predictor. Delaney and Morrison [18] presentcomputational simulation results of a new DEM breakage model for an industrial cone crusher.The model incorporates non-spherical particles represented as superquadrics, which are broken basedon the compressive energy at a contact into non-spherical progeny particles. Asbjörnsson [1] presenteda wear function with data obtained from an actual crusher operating at gradually increasing CSS.Therefore, it is very meaningful to further study the effect of liner wear on cone crusher performance.

The main objective of the present work is to investigate how the liner wear affects cone crusherperformance using the crushing plant test. A model relating a coefficient of material hardness,compression ratio and particle size distribution to a corresponding pressure is presented. Then,combining the empirical model for predicting liner wear with time parameter, a new wear predictionmodel is proposed. The model is important for predicting cone crusher performance along with linerwear. This work can be used for improving cone crusher performance. In addition, this paper describesa method for modelling cone crusher performance along with liner wear. The results of crushing planttest are compared with the corresponding results from the prediction.

2. Experimental Setup

In order to improve cone crusher output, the modern cone crusher chamber should guarantee thatthe rock in the crushing chamber be crushed by interparticle breakage and flow through the chambervia free falling [6,19,20]. In the process of crushing, rock material enters the crushing chamber andkeeps falling until it reaches the choke level. As the mantle moves away from the concave, the rockmaterial becomes loose and falls again. Then rock material meets the mantle and is pushed against theconcave by the mantle. The rock material is squeezed and crushed between the liners. After severalcycles, the crushed rock material falls out of the crushing chamber. All traces of the particle in thecrushing chamber can be obtained. Based on kinematics of free falling, the chamber was studiedby dividing it into several crushing zones [21]. To obtain the pressure model, the correspondingexperiments were designed and performed with a RMT-150B rock mechanics testing system (ChineseAcademy of Sciences, Wuhan, China). In Figure 1, the RMT-150B rock mechanics testing system,a cylindrical container (Chinese Academy of Sciences, Wuhan, China) and a standard sieve (ZhejiangYingchao Instrument Co. Ltd., Shaoxing, China) are shown. The rock material (Shanghai JiansheLuqiao Machinery Co. Ltd., Shanghai, China) was compressed in the cylindrical container. The heightand the diameter of the container were 110 mm and 150 mm. The particle size distribution wasmeasured with the standard sieve. The experiment was designed to simulate the conditions to whicha volume of material was subjected in a real crushing chamber using the RMT-150B rock mechanicstesting system, which can crush rock material according to the experimental parameters includingcompression velocity and compression ratio. The control system of the RMT-150B rock mechanicstesting system is capable of recording compression pressure and stroke data.

Appl. Sci. 2016, 6, 404 3 of 13Appl. Sci. 2016, 6, 404  3 of 13 

 

Figure 1. Experimental processes of rock material interpartical breakage. 

The material chosen for the experiment is basalt, granite, iron ore and limestone, taken from a 

quarry in northern China where the Shanghai Jianshe Luqiao Machinery Co. Ltd. has a test plant. The 

first  experiment  in  each  series was performed on a  sized material of  fraction +9.5−31.5 mm. The 

remaining  experiments were done  on  the material  originating  from  preceding  experiments. The 

experiments were carried out as shown in Figure 1. The particle size distribution of the material was 

measured first, then the rock material was piled into the container. The container was shaken so that 

the rock material was distributed in the container evenly and the top surface of rock material was 

flattened.  The  bed  height  was  measured,  then  the  container  was  set  in  the  RMT‐150B.  The 

experimental  parameters  including  compression  velocity  and  compression  ratio were  set. After 

compression the particle size distribution was measured with sieves, and the results were recorded. 

Subsequently, the steps above were repeated until the rock material could not be compressed any 

more. After setting with a specific compression ratio, the compression ratio was changed to another 

given value and the whole process above was repeated. 

In  order  to  study  changes  of  cone  crusher  performance  along with  liner wear,  a  series  of 

experiments were conducted on processing the full scale crushing plant located in Anhui province, 

China. The rock from Anhui province processed on this processing production line is used for road 

and  building.  This  crushing  plant  is  in  a  quarry  that  produces  approximately  1,100,000  tons  of 

aggregate each year, as is shown in Figure 2a,b. The crushing behavior of the quarry is similar to a 

mining  plant.  The  compressive  strength  of  Anshan  stone  (Anhui  Huagang  Quarry  Co.  Ltd., 

Maanshan, China)  is  110–140 MPa.  In  its  third  crushing  stage,  the  plant  produces  high‐quality 

aggregate products,  ranging  in  size  from 0 mm  to 38 mm. The  cone  crushers are equipped with 

medium chambers of VSC54‐C  (Shanghai Veking Heavy  Industry Co. Ltd., Shanghai, China) and 

VSC54‐F (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China), of which the feed size are 19–

215 mm and 13–115 mm, respectively. 

(a)  (b)

Figure 1. Experimental processes of rock material interpartical breakage.

The material chosen for the experiment is basalt, granite, iron ore and limestone, taken from aquarry in northern China where the Shanghai Jianshe Luqiao Machinery Co. Ltd. has a test plant.The first experiment in each series was performed on a sized material of fraction +9.5−31.5 mm.The remaining experiments were done on the material originating from preceding experiments.The experiments were carried out as shown in Figure 1. The particle size distribution of the materialwas measured first, then the rock material was piled into the container. The container was shaken sothat the rock material was distributed in the container evenly and the top surface of rock material wasflattened. The bed height was measured, then the container was set in the RMT-150B. The experimentalparameters including compression velocity and compression ratio were set. After compression theparticle size distribution was measured with sieves, and the results were recorded. Subsequently,the steps above were repeated until the rock material could not be compressed any more. After settingwith a specific compression ratio, the compression ratio was changed to another given value and thewhole process above was repeated.

In order to study changes of cone crusher performance along with liner wear, a series ofexperiments were conducted on processing the full scale crushing plant located in Anhui province,China. The rock from Anhui province processed on this processing production line is used for road andbuilding. This crushing plant is in a quarry that produces approximately 1,100,000 tons of aggregateeach year, as is shown in Figure 2a,b. The crushing behavior of the quarry is similar to a miningplant. The compressive strength of Anshan stone (Anhui Huagang Quarry Co. Ltd., Maanshan, China)is 110–140 MPa. In its third crushing stage, the plant produces high-quality aggregate products,ranging in size from 0 mm to 38 mm. The cone crushers are equipped with medium chambersof VSC54-C (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China) and VSC54-F (ShanghaiVeking Heavy Industry Co. Ltd., Shanghai, China), of which the feed size are 19–215 mm and13–115 mm, respectively.

Specifically, the third crushing stage was studied. This final stage in the crushing plantproduces high quality aggregate. The working time of actual production line is about 15 h per day;the experiment procedure was as follows: Firstly, the CSS was adjusted to initial value. Then,equipment of the plant was started, and rock materials were crushed. In order to compensate for wearand guarantee product quality, the adjustment of the CSS was implemented manually in 45 h intervalsand took approximately 8–10 min. From 9:00 a.m. to 6:00 p.m., the totals of 27 samples were takenevery 2 h, of which 4 samples were taken before the material crushing. The data of wear was obtainedover time.

Appl. Sci. 2016, 6, 404 4 of 13

Appl. Sci. 2016, 6, 404  3 of 13 

 

Figure 1. Experimental processes of rock material interpartical breakage. 

The material chosen for the experiment is basalt, granite, iron ore and limestone, taken from a 

quarry in northern China where the Shanghai Jianshe Luqiao Machinery Co. Ltd. has a test plant. The 

first  experiment  in  each  series was performed on a  sized material of  fraction +9.5−31.5 mm. The 

remaining  experiments were done  on  the material  originating  from  preceding  experiments. The 

experiments were carried out as shown in Figure 1. The particle size distribution of the material was 

measured first, then the rock material was piled into the container. The container was shaken so that 

the rock material was distributed in the container evenly and the top surface of rock material was 

flattened.  The  bed  height  was  measured,  then  the  container  was  set  in  the  RMT‐150B.  The 

experimental  parameters  including  compression  velocity  and  compression  ratio were  set. After 

compression the particle size distribution was measured with sieves, and the results were recorded. 

Subsequently, the steps above were repeated until the rock material could not be compressed any 

more. After setting with a specific compression ratio, the compression ratio was changed to another 

given value and the whole process above was repeated. 

In  order  to  study  changes  of  cone  crusher  performance  along with  liner wear,  a  series  of 

experiments were conducted on processing the full scale crushing plant located in Anhui province, 

China. The rock from Anhui province processed on this processing production line is used for road 

and  building.  This  crushing  plant  is  in  a  quarry  that  produces  approximately  1,100,000  tons  of 

aggregate each year, as is shown in Figure 2a,b. The crushing behavior of the quarry is similar to a 

mining  plant.  The  compressive  strength  of  Anshan  stone  (Anhui  Huagang  Quarry  Co.  Ltd., 

Maanshan, China)  is  110–140 MPa.  In  its  third  crushing  stage,  the  plant  produces  high‐quality 

aggregate products,  ranging  in  size  from 0 mm  to 38 mm. The  cone  crushers are equipped with 

medium chambers of VSC54‐C  (Shanghai Veking Heavy  Industry Co. Ltd., Shanghai, China) and 

VSC54‐F (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China), of which the feed size are 19–

215 mm and 13–115 mm, respectively. 

(a)  (b)

Figure 2. (a) Typical flow sheet of crushing plant; (b) Typical flow sheet of two-stage aggregatesproduction plant in China.

It is essential to minimize the production of particles sized 0 mm to 4.15 mm and to produce asmany cubic particles as possible for all fraction sizes larger than 4.15 mm. Until recently, the flakinessindex measured the flakiness by means of the Chinese standard T0312-2005. The flakiness index refersto the percentage of flaky particles, which is used to evaluate the particle shape.

3. Theoretical Models and Results

The crushing process was simulated in cone crusher chamber using the laboratory compressioncrushing experiment. After compression, the particle size distribution was measured with sieves.The results were recorded. Subsequently, the pressure data of materials related to different compressionratio and particle size distribution were obtained from the experiment. These experimental data can beused for the development of pressure model and wear models.

3.1. Pressure Model

The pressure model is supposed to describe the relationship between the pressure and thekey factors including compression ratio and particle size distribution. The compression ratio ε isused to describe how much the rock material is compressed. Its value can be calculated usingEquation (1). The size distribution index σ represents the particle size distribution of rock material.It can be calculated using Equations (2) and (3),

ε = s/b, (1)

d =m

∑j=1

wjdj, (2)

σ = [m

∑j=1

wj(dj − d)2]

12

/d, (3)

where s is the stroke of the RMT-150B rock mechanics testing system and its unit is mm, b is the heightof rock material in the container and its unit is mm, wj is the mass percentage of the material in sizerange j, dj is the average size of the material in size range j and its unit is mm, m is the total number ofsize ranges, and d is the average size of all material and its unit is mm. Generally, for cone crushersthe value range of compression ratio is from 0 to 0.4 and the value range of size distribution index isfrom 0 to 1.

Appl. Sci. 2016, 6, 404 5 of 13

After every compression experiment the material is sieved. The pressure data related to differentcompression ratio and particle size distribution can be obtained from the experiments as shown inTable 1.

As the three variable factors can affect values of the compressive pressure, the relationship ofmaterials between the pressure and the compressive ratio is analyzed qualitatively to observe thechange trend of the pressure with the compressive ratio and compression strength. A second orderpolynomial is fitted to the test data in Table 1. These fitting curves are obtained as shown in Figure 3,and relationship between pressure of single factor multi-material crushing and compressive ratio canbe analyzed as shown in Figure 3.

Table 1. Results of crushing experiments.

Granite Basalt

TestNumber

CompressiveRatio (s/b)

SizeDistribution

Index (σ)Compression

Pressure (MPa)Test

NumberCompressive

Ratio (s/b)Size

DistributionIndex (σ)

CompressionPressure (MPa)

1 0.050 0.2621 2.2476 1 0.050 0.2696 3.22482 0.050 0.3324 2.5687 2 0.050 0.2533 3.39243 0.050 0.2577 2.5268 3 0.050 0.2387 3.23884 0.100 0.3701 5.6819 4 0.100 0.3219 7.00815 0.100 0.3777 5.3329 5 0.100 0.3094 6.42186 0.100 0.3844 5.9890 6 0.100 0.3107 6.74287 0.150 0.4253 9.2278 7 0.150 0.4498 13.23448 0.150 0.4478 10.0933 8 0.150 0.4383 10.76349 0.200 0.5408 14.0301 9 0.200 0.5308 14.1000

10 0.200 0.5564 13.9464 10 0.200 0.5208 15.607711 0.200 0.5429 14.0720 11 0.200 0.5312 17.659812 0.250 0.6127 20.5915 12 0.250 0.6801 32.890613 0.300 0.6619 38.2932 13 0.300 0.6852 43.905314 0.300 0.6893 40.2058 14 0.300 0.7567 46.9207

Iron Ore Limestone

TestNumber

CompressiveRatio (s/b)

SizeDistribution

Index (σ)Compression

Pressure (MPa)Test

NumberCompressive

Ratio (s/b)Size

DistributionIndex (σ)

CompressionPressure (MPa)

1 0.050 0.2793 1.2704 1 0.050 0.0263 1.75902 0.050 0.2944 1.8009 2 0.080 0.0263 3.08523 0.050 0.3275 1.8148 3 0.097 0.1461 3.09924 0.100 0.3231 3.7414 4 0.100 0.0263 3.15505 0.100 0.3618 3.6855 5 0.140 0.0263 3.49016 0.100 0.4595 3.5878 6 0.143 0.2677 4.53717 0.150 0.4141 6.7847 7 0.154 0.2678 6.35208 0.150 0.5427 7.5665 8 0.180 0.0263 6.21249 0.150 0.6503 9.9398 9 0.186 0.4103 8.2785

10 0.200 0.4903 9.5349 10 0.189 0.2104 6.449711 0.200 0.6561 13.1646 11 0.221 0.4623 13.764912 0.250 0.5483 13.7091 12 0.260 0.3013 14.197713 0.250 0.7064 14.1558 13 0.271 0.4987 15.886914 0.300 0.6490 26.1198 14 0.281 0.4763 16.1191

s: stroke of the RMT-150B system; b: height of rock material in the container.

Appl. Sci. 2016, 6, 404  5 of 13 

and relationship between pressure of single factor multi‐material crushing and compressive ratio can 

be analyzed as shown in Figure 3. 

Table 1. Results of crushing experiments. 

Granite  Basalt 

Test 

Number 

Compressive 

Ratio (s/b) 

Size 

Distribution 

Index () 

Compression 

Pressure (MPa)

Test 

Number

Compressive 

Ratio (s/b) 

Size 

Distribution 

Index () 

Compression 

Pressure (MPa)

1  0.050  0.2621  2.2476  1  0.050  0.2696  3.2248 

2  0.050  0.3324  2.5687  2  0.050  0.2533  3.3924 

3  0.050  0.2577  2.5268  3  0.050  0.2387  3.2388 

4  0.100  0.3701  5.6819  4  0.100  0.3219  7.0081 

5  0.100  0.3777  5.3329  5  0.100  0.3094  6.4218 

6  0.100  0.3844  5.9890  6  0.100  0.3107  6.7428 

7  0.150  0.4253  9.2278  7  0.150  0.4498  13.2344 

8  0.150  0.4478  10.0933  8  0.150  0.4383  10.7634 

9  0.200  0.5408  14.0301  9  0.200  0.5308  14.1000 

10  0.200  0.5564  13.9464  10  0.200  0.5208  15.6077 

11  0.200  0.5429  14.0720  11  0.200  0.5312  17.6598 

12  0.250  0.6127  20.5915  12  0.250  0.6801  32.8906 

13  0.300  0.6619  38.2932  13  0.300  0.6852  43.9053 

14  0.300  0.6893  40.2058  14  0.300  0.7567  46.9207 

Iron Ore  Limestone 

Test 

Number 

Compressive   

Ratio (s/b) 

Size 

Distribution 

Index () 

Compression 

Pressure (MPa)

Test 

Number

Compressive 

Ratio (s/b) 

Size 

Distribution 

Index () 

Compression 

Pressure (MPa)

1  0.050  0.2793  1.2704  1  0.050  0.0263  1.7590 

2  0.050  0.2944  1.8009  2  0.080  0.0263  3.0852 

3  0.050  0.3275  1.8148  3  0.097  0.1461  3.0992 

4  0.100  0.3231  3.7414  4  0.100  0.0263  3.1550 

5  0.100  0.3618  3.6855  5  0.140  0.0263  3.4901 

6  0.100  0.4595  3.5878  6  0.143  0.2677  4.5371 

7  0.150  0.4141  6.7847  7  0.154  0.2678  6.3520 

8  0.150  0.5427  7.5665  8  0.180  0.0263  6.2124 

9  0.150  0.6503  9.9398  9  0.186  0.4103  8.2785 

10  0.200  0.4903  9.5349  10  0.189  0.2104  6.4497 

11  0.200  0.6561  13.1646  11  0.221  0.4623  13.7649 

12  0.250  0.5483  13.7091  12  0.260  0.3013  14.1977 

13  0.250  0.7064  14.1558  13  0.271  0.4987  15.8869 

14  0.300  0.6490  26.1198  14  0.281  0.4763  16.1191 

s: stroke of the RMT‐150B system; b: height of rock material in the container. 

 

Figure 3. Relationship between pressure of single  factor multi‐material crushing and compressive 

ratio. 

Figure 3. Relationship between pressure of single factor multi-material crushing and compressive ratio.

Appl. Sci. 2016, 6, 404 6 of 13

The crushing pressure increases with the increase of compressive ratio in the compressionexperiment. When the compression ratio is constant, pressure value of basalt is the largest of the fourmaterials. The largest compression strength of crushing materials is basalt in the construction industry,and the value of compression strength of basalt is less than 350 MPa. Thereby, the relative coefficientof material hardness is determined by Equation (4).

λ =K

350, (4)

where K is compression strength and its unit is MPa, λ is a relative coefficient of material hardness.The scatter plots of pressure are obtained using the experimental data, and it is analyzed that

the relationship between these parameters is nonlinear. The relationship between pressure and sizedistribution index is exponential change with the increase of compressive ratio. Therefore, based onnonlinear regression analysis of experimental data, the equation of pressure model was determinedand the coefficients of the model were identified as shown in Equation (5).

F =(

a1λ2 + a2

)exp (a3ε+ a4σ+ a5), (5)

where a1 = 1.0378, a2 = 0.3071, a3 = 7.8752, a4 = 0.8891, a5 = 1.1117.The evaluation index of the fitting model is R-square and Adjusted R-square. The values of

R-square and Adjusted R-square are 0.9871 and 0.9851 respectively, which shows that the model canbetter fit the experimental data. Four kinds of material pressure model are drawn respectively asshown in Figure 4.

Appl. Sci. 2016, 6, 404  6 of 13 

The  crushing  pressure  increases with  the  increase  of  compressive  ratio  in  the  compression 

experiment. When the compression ratio is constant, pressure value of basalt is the largest of the four 

materials.  The  largest  compression  strength  of  crushing materials  is  basalt  in  the  construction 

industry, and the value of compression strength of basalt is less than 350 MPa. Thereby, the relative 

coefficient of material hardness is determined by Equation (4). 

λ=350

K,  (4)

where K is compression strength and its unit is MPa, λ is a relative coefficient of material hardness. 

The scatter plots of pressure are obtained using the experimental data, and it is analyzed that 

the relationship between these parameters is nonlinear. The relationship between pressure and size 

distribution index is exponential change with the increase of compressive ratio. Therefore, based on 

nonlinear regression analysis of experimental data, the equation of pressure model was determined 

and the coefficients of the model were identified as shown in Equation (5). 

λ exp ε σ ,  (5)

where a1 = 1.0378, a2 = 0.3071, a3 = 7.8752, a4 = 0.8891, a5 = 1.1117. 

The evaluation index of the fitting model is R‐square and Adjusted R‐square. The values of R‐

square and Adjusted R‐square are 0.9871 and 0.9851 respectively, which shows that the model can 

better  fit  the experimental data. Four kinds of material pressure model are drawn respectively as 

shown in Figure 4. 

 

Figure 4. Multi‐material crushing pressure model. 

Based on Figure 4, it can be concluded that the pressure will increase, while the size distribution 

index is increasing. That is because the smaller particles move into the interspace of bigger particles 

and protect them from getting crushed. 

3.2. Wear Model 

In compressive crushing, rock particles of various sizes are squeezed and crushed against the 

liner of the crushing chamber. The geometry of the crushing chamber is changed due to the wear of 

crusher liner, which is related with the types of rock material and the pressure distribution on the 

liner. In order to indicate the worn crusher liner influence on the product size distribution and shape 

change, it is necessary to know the previous research on wear model. 

Lindqvist and Evertsson  [15] developed a wear model  to predict  the worn geometry of cone 

crushers. In the model, it is proposed that wear is proportional to the maximum average pressure 

Figure 4. Multi-material crushing pressure model.

Based on Figure 4, it can be concluded that the pressure will increase, while the size distributionindex is increasing. That is because the smaller particles move into the interspace of bigger particlesand protect them from getting crushed.

3.2. Wear Model

In compressive crushing, rock particles of various sizes are squeezed and crushed against theliner of the crushing chamber. The geometry of the crushing chamber is changed due to the wear ofcrusher liner, which is related with the types of rock material and the pressure distribution on the liner.In order to indicate the worn crusher liner influence on the product size distribution and shape change,it is necessary to know the previous research on wear model.

Lindqvist and Evertsson [15] developed a wear model to predict the worn geometry of conecrushers. In the model, it is proposed that wear is proportional to the maximum average pressure

Appl. Sci. 2016, 6, 404 7 of 13

which occurs during the crushing event. The effect of shear forces along the crushing surfaces wasimplemented in the model. The wear is computed according to Equation (6).

∆ω =Fn + QFs

W, (6)

Fn represents normal pressure at the surface, Fs represents shear-stress at the surface, here Q is amodel parameter that scales the effect of the shear force when there is no slip, W is the wear resistancecoefficient depending on the type of liner material and rock, and ∆ω is expressed in mm, pressurein MPa.

However, Equation (6) does not contain a time variable. Actually the liner wear is affected bysome other factors, such as working time and the pressure of crushing zones. The liner wear not onlyconcentrates near the point of the CSS but also distributes the surface of the liner.

In order to solve this problem, a new model of liner wear is proposed, as shown in Equation (7):

∆ωi(t) =Fn,i + QFs,i

Fn,p + QFs,p∗ (µ ∗

t∫0

m(t)dt), (7)

where ∆ωi(t) represents the wear loss of crushing zone i in t hours and its unit is mm, t represents theworking time and its unit is hour , Fn,i represents the normal pressure of crushing zone i, Fs,i representsthe shear-stress of crushing zone i, Fn,p represents the normal pressure near the point of the CSS, Fs,p

represents the shear-stress near the point of the CSS, µ represents the wear rate depending on theamount of crushed material per hour, m(t) represents the crushed product and its unit is kg, pressurein MPa.

The Equation (7) is obtained combining the empirical model for predicting liner wear with thetime parameter. The new model of liner wear contains the productivity variable which is a function oftime variable. Therefore, this model can calculate liner wear with the crusher working, which providesa theoretical model for a significant improvement of the prediction of cone crusher performanceconsidering liner wear.

3.3. Product Quality Model Considering Liner Wear

In mineral engineering, particle size and shape are two key factors that reflect the quality ofproduct [22]. In previous research, the process of the material flowing through the crushing chambercould be modeled as a series of successive crushing events [20]. On the basis of this research and thewear model, equations were formulated for PSD as the liner wear. Size distribution of the final productcan be described by Equations (8) and (9), as shown in:

P(t) =K0

∏i=1

[Bi(t)Si(t) + [1 − Si(t)]]P(1) (t) , (8)

εi(t) =si

bi + ∆ωi (t), (9)

where P(t) is the product size distribution of t time, K0 is the total number of all crushing zones,Si(t) is the selection function and Bi(t) is the breakage function. Si(t) and Bi(t) can be respectivelyobtained by relevant compression experiments. εi(t) is the compression ratio in the crushing zone i,which describes how much the rock material is compressed in the crushing zone i over time.

Piston and die tests were performed to obtain selection function and breakage function [12,23],which were determined by the compression ratio. The compression ratio in crushing zone could becalculated with parameters of the cone crusher structure and wear model. The basalt was used tostudy the selection function Si and breakage function Bi. The selection function model and breakagefunction model was developed, as shown in Equations (10) to (12):

Appl. Sci. 2016, 6, 404 8 of 13

Si(ε(t)i,σi) = 15.3917(ε(t)i)2σi

2 − 4.1471(ε(t)i)2σi − 3.5993(ε(t)i)

2 − 16.2863(ε(t)i)σi2

+4.3881(ε(t)i)σi + 3.8085(ε(t)i) + 0.1613σi

2 − 0.0435σi − 0.0377, (10)

Bi(xN , ε(t)i) = (1 − (0.09545 + 0.4137ε(t)i))x33.62+4.635ε(t)iNi

+(0.09545 + 0.4137((ε(t))i))xNi, (11)

xN =log2(x/xmin)

log2(x0/xmin), (12)

where xmin is the minimum particle diameter of the feed material, x0 is the maximum particle diameterof the feed material, x is the particle diameter needed to solve the relevant weight percentage undersizepassing, and the units of these variables are mm.

The model for particle shape prediction was firstly described by Bengtsson and Evertsson [24].Dong and Fan [25,26] developed a model able to directly calculate the mass percentage of flakinessin the product. The particle shape is also affected by liner wear. Thus, a model of particle shape isdeveloped considering liner wear as shown in Equations (13) and (14):

FITOTAL =n

∑j=1

PjFIj

(F, CSS (t) , dj

), (13)

CSS(t) = CSS + ∆ωp(t) (14)

In this constitutive equation Pj is a component of the vector which represents the proportion ofparticles in the size range j compared to the total product, and n is the total number of size classes.d is the average size of the particles in the size range j, and F is the average particle size of the feed,FITOTAL is just the percentage of flakiness in the total product.

Based on the cone crusher design, the maximum compression ratio in each crushing zone wascalculated and the compression ratio and size distribution index entered into the pressure model, thus,the pressure distribution in the first crushing zone could be achieved. Then, the pressure distributionin each crushing zone and working time of crusher was entered into the wear model, and the linerwear of crusher was emulated. The product size distribution and flakiness index considering linerwear could be obtained by the calculation of models established previously; see Equations (7) to (14).The PSD and particle shape could be estimated from the parameters of data obtained. In this process,it is assumed that the screening operation is ideal, which is a reasonable for assumption of the datanormally used in the test. Therefore, cone crusher performance considering liner wear can be predictedusing the method for calculating the PSD and particle shape.

4. Discussion

A two-variable selection function model and breakage function model were given inEquations (10) to (12) with t as parameter, and constants can be fitted to the experimental data.In order to investigate the effect of compression ratio and size distribution index on the change of therock size in size reduction process, numerical calculation analysis of the selection function model andbreakage function model are implemented. In Figure 5, it is obvious that the curvature of the selectionfunction decreases with increase of size distribution index.

This means that the particle size is of more uneven distribution with an increase of size distributionindex, which will result in a coarse particle not easily broken due to the protection of the tiny particlein the breakage of rock. In Figure 6, a curve denotes the weight percentage of undersized particlespassing a certain particle size, which corresponds to representation of parameters in the breakagefunction and how they are affected by the compression ratio. The size reduction of particles is greatlyimproved when compression ratio is larger.

Appl. Sci. 2016, 6, 404 9 of 13Appl. Sci. 2016, 6, 404  9 of 13 

 

Figure 5. Selection function model. 

 

Figure 6. Breakage function model. 

In order to verify the validity and reliability of the wear model in this work, VSC54‐F, a Chinese 

cone crusher (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China), is selected as an example 

and the relevant models are produced, as shown in Equations (7) to (14). According to the test, the 

relevant test data are obtained, as shown in Figures 7 and 8. 

Typical results from fitting curves of the test data in 45 h are presented in Figures 7 and 8, which 

demonstrate  the size distribution curve and  flakiness  index curve movement over  time, and cone 

crusher performance changes over time. It should be noted that a change moves the size distribution 

curves along the vertical, while a change moves the flakiness index curve along the horizontal axis, 

as shown in Figures 7 and 8. 

Figure 5. Selection function model.

Appl. Sci. 2016, 6, 404  9 of 13 

 

Figure 5. Selection function model. 

 

Figure 6. Breakage function model. 

In order to verify the validity and reliability of the wear model in this work, VSC54‐F, a Chinese 

cone crusher (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China), is selected as an example 

and the relevant models are produced, as shown in Equations (7) to (14). According to the test, the 

relevant test data are obtained, as shown in Figures 7 and 8. 

Typical results from fitting curves of the test data in 45 h are presented in Figures 7 and 8, which 

demonstrate  the size distribution curve and  flakiness  index curve movement over  time, and cone 

crusher performance changes over time. It should be noted that a change moves the size distribution 

curves along the vertical, while a change moves the flakiness index curve along the horizontal axis, 

as shown in Figures 7 and 8. 

Figure 6. Breakage function model.

In order to verify the validity and reliability of the wear model in this work, VSC54-F, a Chinesecone crusher (Shanghai Veking Heavy Industry Co. Ltd., Shanghai, China), is selected as an exampleand the relevant models are produced, as shown in Equations (7) to (14). According to the test,the relevant test data are obtained, as shown in Figures 7 and 8.Appl. Sci. 2016, 6, 404  10 of 13 

 

Figure 7. Worn liner causes the size distribution curve to change over time. 

 

Figure 8. Worn liner causes the flakiness curve to change over time. 

The prediction data of product size distribution are obtained by Equations (7) to (14), and the 

measured data are obtained from field test, as shown in Table 2.   

Table 2. Comparison between measured and predicted results of the particle size distribution (PSD). 

Size (mm)  Measured (%) Considering Liner Wear Without Considering Liner Wear

Predicted (%) Error (%) Predicted (%) Error (%) 

+37.5  88.2  86.4  2.0  90.8  −3.0 

+31.5−37.5  65.5  65.1  0.6  71.6  −9.3 

−26.5+31.5  49.0  49.8  −1.6  57.1  −16.5 

+19−26.5  28.5  30.8  −8.1  38.0  −33.3 

+16−19  21.6  24.6  −13.9  31.3  −44.9 

+9.5−16  9.6  10.8  −12.5  14.1  −46.9 

The measured data of flakiness index are obtained by Equations (7) to (14), and the measured 

data are obtained from field test, as shown in Table 3. 

 

Figure 7. Worn liner causes the size distribution curve to change over time.

Appl. Sci. 2016, 6, 404 10 of 13

Appl. Sci. 2016, 6, 404  10 of 13 

 

Figure 7. Worn liner causes the size distribution curve to change over time. 

 

Figure 8. Worn liner causes the flakiness curve to change over time. 

The prediction data of product size distribution are obtained by Equations (7) to (14), and the 

measured data are obtained from field test, as shown in Table 2.   

Table 2. Comparison between measured and predicted results of the particle size distribution (PSD). 

Size (mm)  Measured (%) Considering Liner Wear Without Considering Liner Wear

Predicted (%) Error (%) Predicted (%) Error (%) 

+37.5  88.2  86.4  2.0  90.8  −3.0 

+31.5−37.5  65.5  65.1  0.6  71.6  −9.3 

−26.5+31.5  49.0  49.8  −1.6  57.1  −16.5 

+19−26.5  28.5  30.8  −8.1  38.0  −33.3 

+16−19  21.6  24.6  −13.9  31.3  −44.9 

+9.5−16  9.6  10.8  −12.5  14.1  −46.9 

The measured data of flakiness index are obtained by Equations (7) to (14), and the measured 

data are obtained from field test, as shown in Table 3. 

 

Figure 8. Worn liner causes the flakiness curve to change over time.

Typical results from fitting curves of the test data in 45 h are presented in Figures 7 and 8,which demonstrate the size distribution curve and flakiness index curve movement over time,and cone crusher performance changes over time. It should be noted that a change moves thesize distribution curves along the vertical, while a change moves the flakiness index curve along thehorizontal axis, as shown in Figures 7 and 8.

The prediction data of product size distribution are obtained by Equations (7) to (14), and themeasured data are obtained from field test, as shown in Table 2.

Table 2. Comparison between measured and predicted results of the particle size distribution (PSD).

Size (mm) Measured (%)Considering Liner Wear Without Considering Liner Wear

Predicted (%) Error (%) Predicted (%) Error (%)

+37.5 88.2 86.4 2.0 90.8 −3.0+31.5−37.5 65.5 65.1 0.6 71.6 −9.3−26.5+31.5 49.0 49.8 −1.6 57.1 −16.5+19−26.5 28.5 30.8 −8.1 38.0 −33.3+16−19 21.6 24.6 −13.9 31.3 −44.9+9.5−16 9.6 10.8 −12.5 14.1 −46.9

The measured data of flakiness index are obtained by Equations (7) to (14), and the measureddata are obtained from field test, as shown in Table 3.

Table 3. Comparison between measured and predicted results of the flakiness index.

Size (mm) Measured (%)Considering Liner Wear Without Considering Liner Wear

Predicted (%) Error (%) Predicted (%) Error (%)

+31.5 19.5 19.9 −2.1 23.9 −22.6+26.5−31.5 20.4 19.1 6.4 18.8 7.8+19.0−26.5 31.4 28.7 8.6 24.7 21.3+16.0−19 39.2 36.3 7.4 31.6 19.4+9.5−16 62.4 59.9 4.0 55.4 11.2

Tables 2 and 3 summarize the results from the tests and compute at 45 h under normal feedconditions. They show the comparison between computed results of product size distribution and

Appl. Sci. 2016, 6, 404 11 of 13

flakiness index considering liner wear and those without considering liner wear. The errors could becalculated to analyze the accuracy of the results.

By analyzing the measured and predicted results of size distribution and flakiness index,the errors in the predicted values can be calculated, as illustrated in Tables 2 and 3. Table 2 summarizesthe results from the tests and predicted values of the PSD. It shows that the absolute values of errors inpredicted values of the PSD without considering liner wear are less than 46.9% and those consideringliner wear are less than 13.9%, with respect to the test values. This demonstrates the predicted valuesconsidering liner wear are better than those without considering liner wear. Table 3 shows measuredand predicted results of the flakiness index of the product. Compared to results without consideringwear, the predicted results considering wear shows significant improvement in representing themeasured data, showing an only 8.6% maximum deviation from the measured results. Therefore,cone crusher performance considering liner wear is predicted using the proposed new model in thispaper, which can improve the accuracy of the prediction results.

5. Conclusions

In order to compute the product size distribution and flakiness index along with liner wear, testsare performed both to collect data and to validate the method in this paper. A wear model for theprediction of cone crusher performance is presented, which is able to be improved in order to betterrepresent the actual process. By considering the wear model for prediction of cone crusher performancein the crushing process, the agreement between predicted and measured crusher performance issignificantly improved. This work will be useful for future dynamical optimization of the crushingplant, which can serve to increase plant insensitivity to variations, such as wear and malfunctions.These tests give a clear indication that there is a correlation between liner wear and a change of theflakiness index and PSD. However, it is not possible to say that this change is only related to linerwear. There are several other factors, such as rock properties and feed size distribution that alsocould promote the change of the flakiness index and PSD. The purpose of this paper is not only tocreate a general model for all crushers but also to obtain a way to better represent the actual sizereduction process.

Future work should focus on utilization of the model for dynamic plant performance optimizationand evaluating the effects of plant design.

Acknowledgments: The work was supported by Doctoral Program of Higher Education Research Fund(Grant No. 20090073110038).

Author Contributions: Yanjun Ma, Xiumin Fan and Qichang He designed the experiments; Yanjun Ma performedthe experiments; Yanjun Ma and Qichang He analyzed the data; Xiumin Fan contributed anlysis tools; Yanjun Mawrote the paper.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix

The following symbols are used in this manuscript:

Symbols Symbolic Meaning Units

s Stroke of the RMT-150B system mmb Height of rock material in the container mmε Compression ratio -σ Size distribution index -wj Mass percentage of the material in size range j -dj Average size of the material in size range j -m Total number of size ranges -d Average size of all material mmK Compression strength MPa

Appl. Sci. 2016, 6, 404 12 of 13

Symbols Symbolic Meaning Units

λ A relative coefficient of material hardness -F Compression pressure MPaFn Normal pressure at the surface MPaFs Shear-stress at the surface MPaQ A model parameter -W Wear resistance coefficient -

∆ω Liner wear mm∆ωi(t) Wear loss of crushing zone i in t hours mm

t Working time hourFn,i Normal pressure of crushing zone i MPaFs,i Shear-stress of crushing zone i MPaFn,p Normal pressure near the point of the CSS MPaFs,p Shear-stress near the point of the CSS MPaµ Wear rate -

m(t) Crushed product kgP(t) Product size distribution of t time -K0 Total number of all crushing zones -

Si(t) Selection function -Bi(t) Breakage function -εi(t) Compression ratio in the crushing zone i -xmin Minimum particle diameter of the feed material mmx0 Maximum particle diameter of the feed material mm

xParticle diameter needed to solve the relevant weightpercentage undersize passing

mm

PjA component of the vector which represents theproportion of particles in the size range j

-

n Total number of size classes -F Average particle size of the feed mm

FITOTAL Percentage of flakiness -

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